Emily Riehl: So I'm a category theorist, and every category theorist's favorite theorem is the Yoneda Lemma. It says that a mathematical object of some kind is uniquely determined by the relationships that it has to all other objects of the same type. In fact, it's uniquely characterized in two different ways. You can either look at maps from the object you're trying to understand or maps to the object you're trying to understand, and either way suffices to determine it.

This is an amazing theorem. There's a joke in category theory that all proofs are the Yoneda Lemma. I mean, all proofs reduce to the Yoneda Lemma.

The reason I don't want to talk about it today is two-fold. Number one, the discussion might sound a little more philosophical than mathematical, because one thing that the Yoneda Lemma does is that it orients the philosophy of category theory. Secondly, there's this wonderful experience you have as a student when you see the Yoneda Lemma for the first time, because the statement you'll probably see is not the one I just described but sort of a weirder one involving natural transformations from representable functors, and you see it, and you're like, "Okay, I guess that's plausible, but why on earth would anyone care about this?" And then it sort of dawns on you, over however many years, in my case, why it's such a profound and useful observation. So I don't want to ruin that experience for anybody.